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a, {w 3 EE 411, 20080227 Quiz 6 15650 Name: ____ EID: (Q1) If the switch in the figure below has been closed for a long time before t = O , but is
opened at t = 0 determine: (a) the characteristic equation of the circuit,
(b) ixand V]: for t >0. (a) Let V = capacitor voltage and i : inductor current. At t = 0, the switch is
closed and the circuit has reached steady—state. v(O‘) = 16V and i(0—) = 16/8 = 2A
At t = 0+, the'switch'is Open but, v(0+) ="16 and i(0+)’ = 2.
We now have a sourcefree RLC circuit.
R 8+12 = ZOohms, L 2 1H, C = 4mF.
OL = R/(2L) = (20)/(2X1) = 10 (00: 1/\/LC = 1/ 1X(1/36) = 6 Since on > (no, we have a overdarnped response. 81,2 = —oci1/0c2 —co(2, = ~18,2 Thus, the characteristic equation is (s + 2)(s + 18) = O or 52 + 20s +36 = 0.
(b) i(t) = [Ae‘2‘+Be'13‘] and i(O) = 2 = A+B (1) To get di(0)/dt, consider the circuit below at t 2 0+. EE 411, 20030227 Quiz 6 ' 15650 Name: EID: 129 39 + V —V(O) + 201(0) + VL(O) = O, which leads to,
~16 +20x2 +VL(O) = 0 or VL(0) = —24
Ldi(O)/dt = VL(O) which gives di(0)/dt = VL(0)/L = —24/1 = ~24 A/s
Hence —24 = —2A— 18B or 12 = A+9B (2)
From (1) and (2), B = 1.25 andA = 0.75
1(1) = [0,7552% 1.25e'18t] = —ix(t) or ix(t) = [0.75e'2t 1.25e'18t] A
Va) = 81(t) = [6e'2‘+ 10e'18‘] A ...
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 Spring '07
 Lee

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